Finite groups with a large normalized sum of element orders

Abstract For a finite group G, let (G) ψ (G) be the sum of the orders of its elements, and define the corresponding normalized sum as ' (G): = (G) / (C|₆|) ψ ′ (G): = ψ (G) / ψ (C | G |), where C|₆| C | G | is the cyclic group of the same order as G. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if ' (G) > ' (D₈) = 1943 ψ ′ (G) > ψ ′ (D 8) = 19 43, then G belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying ' (G) > ' (A₄) = 3177 ψ ′ (G) > ψ ′ (A 4) = 31 77, thereby fully determining the groups covered by the supersolubility criterion of Baniasad Azad and Khosravi Canad. Math. Bull. 65 (2022), 30–38, and thus providing a more complete answer to a corresponding conjecture of Tǎrnǎuceanu.